Supremum of Banach Spaces Let $X$ be a linear space with a family of complete norms $(\| \circ \|_I)_{I \in \mathcal{I}}$ on $X$, i.e. for every $I \in \mathcal{I}$ the tuple $(X,\|\circ\|_I)$ is a Banach space. Now define $$\| x \|_\infty := \sup_{I \in \mathcal{I}} \| x \|_I \in [0,\infty]$$ for every $x \in X$ and $X_\infty = \{ x \in X ~:~ \|x\|_\infty < \infty \}$. It is easy to see that $(X_\infty,\|\circ\|_\infty)$ is a normed space. In particular, $X_\infty$ is a linear space.
Is $(X_\infty,\|\circ\|_\infty)$ a Banach space, i.e. is it complete?
My guess would be negative, but I can't come up with an (counter-)example. 
What I have tried so far: It is easy to see that if $X$ is finite-dimensional and $\mathcal{I}$ is finite, then $(X_\infty,\|\circ\|_\infty)$ is again complete. Therefore I asumme that either I have to find many "incompatible" norms on a finite-dimensional space, or only "a few" norms on a "complicated" space. I would appreciate hints for either direction.
 A: The answer is no.
If $I$ is finite, then $X_\infty=X$ as linear spaces, so I assume $I$ is finite. Using open mapping theorem one can show that if $(X_\infty,\Vert\cdot\Vert_\infty)$ is Banach space then all the spaces $(X_\infty,\Vert\cdot\Vert_i)$ are isomorphic. Indeed, for all $i\in I$ the map
$$
\mathrm{Id}:(X_\infty,\Vert\cdot\Vert_\infty)\to(X_\infty,\Vert\cdot\Vert_i):x\mapsto x
$$
is bounded and bijective, hence isomorphism. Since for all $i\in I$ we have $(X_\infty,\Vert\cdot\Vert_\infty)\cong(X_\infty,\Vert\cdot\Vert_i)$, the spaces $(X_\infty,\Vert\cdot\Vert_i)$ for $i\in I$ are pairwise isomorphic.
Now to construct a counterexample it is sufficient to introduce finitely many non-isomorphic Banach space structures on the same linear space $X$. I'll construct two structures.
Let $(X,\Vert\cdot\Vert)$ be a separable Banach space non-isomorphic to $\ell_2$. For example take $X=(\ell_p,\Vert\cdot\Vert_p)$ with $p\neq 2$. Now consider norm $\Vert\cdot\Vert'$ given in this answer which turns $X$ into a Hilbert space. By construction $(X,\Vert\cdot\Vert)$ and $(X,\Vert\cdot\Vert')$ are not isomorphic. So norms $\Vert\cdot\Vert$ and $\Vert\cdot\Vert'$ gives the desired counterexample.
